Classification of Schmidt-rank-two multipartite unitary gates by singular number

TL;DR

This paper classifies genuine multipartite Schmidt-rank-two unitary gates using a novel singular number metric, providing a detailed parametric description and extending the analysis to three-qubit diagonal gates.

Contribution

It introduces the singular number as a new classification tool for multipartite unitary gates of Schmidt rank two and characterizes three-qubit diagonal unitary gates.

Findings

Singular number ranges from 0 to the maximum for given gates.

Parametric Schmidt decompositions are formulated for each singular number.

Three-qubit diagonal unitary gates have Schmidt rank at most three, with conditions for rank three.

Abstract

The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary gates of Schmidt rank two, by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of Schmidt rank two, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number to classify the unitary gates concerned. The singular number is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of singular number. For each singular number, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and Schmidt-rank-two unitaries. We start with discussing two typical examples of Schmidt rank two, one of which is a fundamental three-qubit unitary gate, i.e., the CCZ gate. Then we characterize the diagonal unitary gates of Schmidt rank greater than two. We show that a three-qubit diagonal unitary gate has Schmidt rank at most three, and present a necessary and sufficient condition for such a unitary gate of Schmidt rank three. This completes the characterization of all genuine three-qubit diagonal unitary gates.